Artin's Primitive Root Conjecture Research Articles

Prime divisors of ℓ-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level ℓ

Let ℓ be any fixed prime number. We define the ℓ-Genocchi numbers by Gn:=ℓ(1−ℓn)Bn, with Bn the n-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime p is ℓ-Genocchi irregular if it divides at least one of the ℓ-Genocchi numbers G2,G4,…,Gp−3, and ℓ-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of ℓ-Genocchi irregular primes in a prescribed arithmetic progression in case ℓ is odd. The case ℓ=2 was already dealt with by Hu et al. (2019) [14]. Using similar methods we study the prime factors of (1−ℓn)B2n/2n and (1+ℓn)B2n/2n. This allows us to estimate the number of primes p≤x for which there exist modulo p Ramanujan-style congruences between the Fourier coefficients of an Eisenstein series and some cusp form of prime level ℓ.

Zigzag polynomials, Artin's conjecture and trinomials

Let Fp denote the finite field of prime order p. We introduce a new natural family of polynomials in Fp[X]. These polynomials first arose from considering faithful representations of the elementary abelian group of order p2 in characteristic p. The polynomials exhibit a number of interesting special properties. For example they satisfy a three term recursion, are closely related to zigzag zero-one sequences and form strong divisibility sequences. These polynomials are shown to be closely connected to the order of appearance of prime numbers in the Fibonacci sequence, Artin's Primitive Root Conjecture, and the factorization of trinomials over finite fields.

Simultaneous insolvability of exponential congruences

We determine a necessary and sufficient condition for the infinitude of primes p such that none of the equations aix≡bi(modp),1≤i≤n, are solvable. We control the insolvability of ax≡b(modp) by power residues for multiplicatively independent a and b, and by divisibilities and, most importantly, parities of orders in multiplicatively dependent cases. We also consider a more general problem concerning divisibilities of orders. The problems are motivated by Artin's primitive root conjecture and its variants.

Artin's primitive root conjecture for function fields revisited

Artin's primitive root conjecture for function fields was proved by Bilharz in his thesis in 1937, conditionally on the proof of the Riemann hypothesis for function fields over finite fields, which was proved later by Weil in 1948. In this paper, we provide a simple proof of Artin's primitive root conjecture for function fields which does not use the Riemann hypothesis for function fields but rather modifies the classical argument of Hadamard and de la Vallée Poussin in their 1896 proof of the prime number theorem.

Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture

We introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-(ir)regularity and is based on Genocchi rather than Bernoulli numbers. We say that an odd prime p is G-irregular if it divides at least one of the Genocchi numbers G2,G4,…,Gp−3, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound x as x tends to infinity. As a byproduct, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root.

Corrigendum The Generalized Artin Primitive Root Conjecture (EUROPEAN J. OF PURE AND APPLIED MATH. Vol. 11, No. 1, 2018, 23-34)

The subset of integers $\mathcal{N}_2= \{ n\in \mathbb{N}:\text{ord}_n(2)=\lambda(n) \}$ in page 24, \cite{CN18}, should be \\ $$\label{eq2-40} \mathcal{N}_u =\left\{ n\in \mathbb{N}:\ord_n(u)=\lambda(n) \text{ and } p \mid n \Rightarrow \ord_p(u)=p-1, \ord_{p^2}(u)=p(p-1) \right\} $$ where $u \ne \pm 1, v^2$.In addition, Lemma 3.4 was corrected. These changes do not affect the main result. The proof of Theorem 1.1 remains the same. The new version of the paper is available at arXiv:1504.00843v9.

Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over $\F_q,$ for many classes of $q$'s. This is a partial solution to a fifty year old open problem.

Long quasi-polycyclic <inline-formula><tex-math id="M1">\begin{document}$t-$\end{document}</tex-math></inline-formula> CIS codes

We study complementary information set codes of length $tn$ and dimension $n$ of order $t$ called ($t-$CIS code for short). Quasi-cyclic (QC) and quasi-twisted (QT) $t$-CIS codes are enumerated by using their concatenated structure. Asymptotic existence results are derived for one-generator and fixed co-index QC and QT codes depending on Artin's primitive root conjecture. This shows that there are infinite families of rate $1/t$ long QC and QT $t$-CIS codes with relative distance satisfying a modified Varshamov-Gilbert bound. Similar results are defined for the new and more general class of quasi-polycyclic codes introduced recently by Berger and Amrani.

Upper bounds for the number of primitive ray class characters with conductor below a given bound

We present upper bounds on certain sums which are related to Artin's primitive root conjecture and are also used in counting ray class characters.

The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank

AbstractLet ψ be a generic Drinfeld module of rank r ≥ 2. We study the first elementary divisor d1,℘ (ψ) of the reduction of ψ modulo a prime ℘, as ℘ varies. In particular, we prove the existence of the density of the primes ℘ for which d1,℘ (ψ) is fixed. For r = 2, we also study the second elementary divisor (the exponent) of the reduction of ψ modulo ℘ and prove that, on average, it has a large norm. Our work is motivated by J.-P. Serre's study of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M. R. Murty.

Artin's primitive root conjecture and a problem of Rohrlich

AbstractLet$\mathbb{K}$be a number field, Γ a finitely generated subgroup of$\mathbb{K}$*, for instance the unit group of$\mathbb{K}$, and κ>0. For an ideal$\mathfrak{a}$of$\mathbb{K}$let indΓ($\mathfrak{a}$]></alt-text></inline-graphic>) denote the multiplicative index of the reduction of Γ in <inline-graphic name="S0305004114000206_inline3"><alt-text><![CDATA[$(\mathcal{O}_\mathbb{K}/\mathfrak{a})$* (whenever it makes sense). For a prime ideal$\mathfrak{p}$of$\mathbb{K}$and a positive integer γ let$\mathcal{I}_\gamma^\kappa(\mathfrak{p})$be the average of${ind}_{\langle a_1,\dots,a_\gamma\rangle}(\mathfrak{p})^\kappa$over all tupels$(a_1,\dots,a_\gamma)\in{(\mathcal{O}_\mathbb{K}/\mathfrak{p})^*}^\gamma$. Motivated by a problem of Rohrlich we prove, partly conditionally on fairly standard hypotheses, lower bounds for$\sum_{\mathcal{N}{\mathfrak{a}\leq x}{ind}_{\Gamma}({\mathfrak{a})^\kappa$and asymptotic formulae for$\sum_{\mathcal{N}\mathfrak{p} \leq x} {\mathcal{I}_{\gamma}^\kappa({\mathfrak{p})$.

Artin's Primitive Root Conjecture – A Survey

Abstract.One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on `elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on `Artin for K-theory of number fields,' and Hester Graves (together with me) on `Artin's conjecture and Euclidean domains.'

Periods of orbits modulo primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α ∈ V ( K ) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin's primitive root conjecture.

On primes in arithmetic progression having a prescribed primitive root.II

Let $a$ and $f$ be coprime positive integers. Let $g$ be an integer. Under the Generalized Riemann Hypothesis (GRH) it follows by a result of H.W. Lenstra that the set of primes $p$ such that $p\equiv a({\rm mod~}f)$ and $g$ is a primitive root modulo $p$ has a natural density. In this note this density is explicitly evaluated with an Euler product as result. This extends a classical result of Hooley (1967) on Artin's primitive root conjecture. Various application are given, for example the integers $g$ and $f$ such that the set of primes $p$ such that $g$ is a primitive root modulo $p$ is equidistributed modulo $f$ is determined (on GRM).

Counting cusps of subgroups of $\mathrm {PSL}_2(\mathcal {O}_K)$

Let K be a number field with r real places and s complex places, and let O K be the ring of integers of K. The quotient [H 2 ] r X [H 3 ] s /PSL 2 (O K ) has h K cusps, where h K is the class number of K. We show that under the assumption of the generalized Riemann hypothesis that if K is not Q or an imaginary quadratic field and if i ∉ K, then PSL 2 (O K ) has infinitely many maximal subgroups with h K cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

Primitive roots in quadratic fields, II

We consider an analogue of Artin's primitive root conjecture for algebraic numbers which are not units in quadratic fields. Given such an algebraic number α, for a rational prime p which is inert in the field, the maximal possible order of α modulo ( p ) is p 2 − 1 . An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. We show that for any choice of 113 algebraic numbers satisfying a certain simple restriction, at least one of the algebraic numbers has order at least p 2 − 1 48 mod ( p ) for infinitely many inert primes p.

PRIMITIVE ROOTS IN QUADRATIC FIELDS

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture.

The correction factor in Artin's primitive root conjecture

En 1927, E. Artin proposait une densite conjecturale pour l'ensemble des nombres premiers p pour lesquels un entier donne g est une racine primitive modulo p. Des calculs effectues en 1957 par D. H. et E. Lehmer ont mis en evidence des ecarts inattendus par rapport a cette densite conjecturale, incitant Artin a y introduire un facteur correctif. La conjecture ainsi modifiee a ete prouvee par C Hooley en 1967 conditionnellement a l'hypothese de Riemann generalisee. Cet article a pour sujet deux developpements recents relatifs a ce facteur correctif. Le premier est de nature historique, et se rapporte a des lettres, recemment decouvertes a Berkeley dans les archives des Lehmer, entre Artin et les Lehmer durant les annees 1957-58. Le second concerne une nouvelle interpretation du facteur correctif, due a H. W. Lenstra, P. Moree et l'auteur, qui peut s'appliquer a de nombreuses generalisations au probleme originel d'Artin.

Artin's primitive root conjecture for quadratic fields

Soit a fixe dans un corps quadratrique K. On note S l'ensemble des nombres premiers p pour lesquels a admet un ordre maximal modulo p. Sous G.R.H., on montre que S a une densite. Nous donnons aussi des conditions necessaires et suffisantes pour que cette densite soit strictement positive.

Prime divisors of linear recurrences and Artin's primitive root conjecture for number fields

Let S be a linear integer recurrent sequence of order k≥3, and define P S as the set of primes that divide at least one term of S. We give a heuristic approach to the problem whether P S has a natural density, and prove that part of our heuristics is correct. Under the assumption of a generalization of Artin’s primitive root conjecture, we find that P S has positive lower density for “generic” sequences S. Some numerical examples are included.